ź-nets and simplex range queries
David Haussler,Emo Welzl +1 more
TLDR
The concept of an ɛ-net of a set of points for an abstract set of ranges is introduced and sufficient conditions that a random sample is an Â-net with any desired probability are given.Abstract:
We demonstrate the existence of data structures for half-space and simplex range queries on finite point sets ind-dimensional space,dÂ?2, with linear storage andO(nÂ?) query time, $$\alpha = \frac{{d(d - 1)}}{{d(d - 1) + 1}} + \gamma for all \gamma > 0$$ .
These bounds are better than those previously published for alldÂ?2. Based on ideas due to Vapnik and Chervonenkis, we introduce the concept of an Â?-net of a set of points for an abstract set of ranges and give sufficient conditions that a random sample is an Â?-net with any desired probability. Using these results, we demonstrate how random samples can be used to build a partition-tree structure that achieves the above query time.read more
Citations
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Diameter, Eccentricities and Distance Oracle Computations on H-Minor Free Graphs and Graphs of Bounded (Distance) Vapnik-Chervonenkis Dimension
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Parallel Randomized Techniques for Some Fundamental Geometric Problems
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Cuttings for disks and axis-aligned rectangles
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Planar point sets determine many pairwise crossing segments
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References
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TL;DR: In this article, the convergence of a stochastic process indexed by a Gaussian process to a certain Gaussian processes indexed by the supremum norm was studied in a Donsker class.
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